Has anybody pointed out that it might be possible to look at this from the point of view of what is best for your DD long term, rather than as a possible way your XH might take something from you?
For instance: presumably before you became entitled to the discount you could both afford the fees 50/50 and you were happy with the situation around holiday care (ie. you wish to have less stress and spend time with DD etc). If you can both still afford a 50/50 split of the revised fees how about suggesting you share this (as long as any tax liability you might pay is addressed so you're not worse off) and each pay the other half into a savings account for your DD future school needs or university?
It is great that one of you could afford to cover the fees if the other fell on hard times but this way you'd be jointly saving to cover such an eventuality.
I have a DD and her father is my X and I think its very easy to get caught in this kind of trap of being convinced in advance that the other party is going to try to hurt you financially. Maybe he'd say - "that's your discount don't worry about it".
I wouldn't try to bring in the childcare thing unless the current arrangement has caused you upset, as it suggests there's underlying resentment about an arrangement you've previously agreed.
You don't have automatic access to his benefits and he doesn't have access to yours. But your relationship sounds amicable and this benefit is really for your DD. If his work offered discounted swimming lessons for children and he let you pay 50% of the full price and took the discount out of his side you might be annoyed..? Or at least feel it would have been nice to have been consulted, even if you wouldn't dream of sharing his benefit.
The true beneficiary here is your DD, and the more you can do to help each other sustain the situation over the long term the better it will be for her. It can't be easy paying private school fees and it's probably not worth falling out over - this should be a great thing for you all!
Life is long, and expensive... parents helping each other to help their child should be an easy choice.
(And congratulations on the new job!)